GCSE Higher Maths Topics by Grade (Edexcel)

3 papers1 non-calculator + 2 calculator

240 marks80 marks per paper

Grades 4–9Higher tier range

100% examNo coursework

Where each grade sits

4Standard pass

5Strong pass

6Sixth form

7A‑Level ready

8High achiever

9Top 1 in 20

Grade 5 typically needs ~30–36% of raw marks on Higher papers; grade 9 needs ~85–90%. Boundaries move slightly each year — see the table below.

Grades 4–5

Rounding & estimation: Round to sig. figs / dec. places; estimate by rounding to 1 s.f. before calculating.
Fractions, decimals & %: Convert fluently between forms; order them; calculate with mixed operations.
Indices (basic): Positive/negative integer powers; laws of indices.
Factors, multiples, primes: Product of prime factors, HCF, LCM via Venn diagrams or listing.
Standard form (basic): Write/interpret numbers in standard form; simple × and ÷.

Grades 6–7

Standard form (non-calc): Multiply/divide without a calculator, including negative indices.
Surds: Simplify √a (e.g. √50 = 5√2); multiply/divide surds.
Fractional & negative indices: Evaluate a^1/2, a⁻², a^2/3; solve equations involving indices.
Recurring decimals: Convert to exact fractions using the algebraic method.
Bounds & error intervals: State upper/lower bounds from a rounded value.

Grades 8–9

Surds (advanced): Rationalise denominators (incl. binomial surds); expand brackets containing surds.
Bounds (compound): Use bounds in multi-step / compound-measure calculations.
Exact calculations: Leave answers in terms of π or surds throughout a multi-step problem.
Growth/decay reasoning: Multi-step compound interest or depreciation combined with algebra or ratio.

Grades 4–5

Expanding & factorising: Single/double brackets; factorise x²+bx+c into (x+p)(x+q).
Solving equations: Linear equations, incl. unknown on both sides, brackets, fractions.
Substitution & formulae: Substitute into expressions; rearrange simple formulae (subject once).
Sequences (linear): Find and use the nth term of a linear sequence.
Straight-line graphs: Plot from a table; find gradient and y-intercept; y = mx + c.
Simultaneous equations: Solve linear–linear pairs by elimination or substitution.

Grades 6–7

Quadratics: Factorise ax²+bx+c (a≠1); solve by factorising or the quadratic formula.
Algebraic fractions (basic): Simplify by cancelling common factors.
Rearranging formulae: Subject appears twice, or requires factorising to isolate it.
Quadratic graphs: Sketch/plot y = x²+bx+c; identify roots, turning point, intercepts.
Inequalities: Solve and represent linear inequalities; graphical regions.
Simultaneous equations: One linear, one quadratic (straightforward cases).

Grades 8–9

Completing the square: Write x²+bx+c as (x+p)²+q; use to find turning point or solve equations.
Algebraic fractions (advanced): Add/subtract with different denominators; solve equations containing them.
Functions: Composite functions fg(x); inverse functions f⁻¹(x).
Iteration: Use an iterative formula x_n+1 = … to find approximate roots.
Sequences (quadratic): Find the nth term of a quadratic sequence.
Graph transformations: f(x)+a, f(x+a), −f(x), f(−x), af(x) — describe and sketch.
Rates of change: Estimate gradient of a curve at a point; area under a curve as a rate/quantity.

Grades 4–5

Ratio: Simplify; share in a given ratio; find one part/whole given the other.
Direct proportion: Unitary method; best-buy / value-for-money comparisons.
Percentages: Percentage increase/decrease; reverse percentage (find the original).
Units & conversions: Convert between metric units; simple speed/distance/time and density.

Grades 6–7

Compound measures: Density, pressure, speed — incl. changing the subject and mixed units.
Inverse proportion: Recognise and solve inverse proportion problems (y ∝ 1/x).
Growth & decay: Compound interest, depreciation, multi-year percentage problems.
Direct/inverse (algebraic): Set up and solve y = kx or y = k/x, find k, then find unknowns.

Grades 8–9

Proportion (non-linear): y = kx², y = kx³, y = k√x, y = k/x² — set up, find k, solve.
Rates of change from graphs: Interpret gradient as a rate (speed, acceleration).
Algebraic proof with ratio: Prove a ratio/proportion relationship holds generally, using algebra.

Grades 4–5

Angle facts: Triangles, quadrilaterals, parallel lines; angle sum of polygons.
Area & perimeter: Rectangles, triangles, trapezia, compound shapes; circles.
Pythagoras' theorem: Find a missing side in a right-angled triangle.
Trigonometry (basic): SOHCAHTOA for one missing side or angle.
Transformations (single): Translate, reflect, rotate, enlarge.
Volume & surface area: Cuboids, prisms, cylinders.

Grades 6–7

Circle theorems (core): Angle at centre = 2× at circumference; angle in semicircle = 90°; tangent–radius.
Sine & cosine rules: Non-right-angled triangles; area = ½ab sin C.
Similar shapes: Length, area, and volume scale factors.
3D Pythagoras & trig: Lengths/angles within cuboids, pyramids, and other solids.
Transformations (combined): Describe a single transformation equivalent to two or more.

Grades 8–9

Circle theorems (all 8): Including alternate segment theorem, cyclic quadrilaterals; formal proofs.
Vectors: Vector arithmetic and geometric proof (e.g. show three points are collinear).
Congruence proofs: Formal proof using SSS, SAS, ASA, RHS.
Complex trigonometry: Multi-step problems combining bearings, sine/cosine rule, 3D reasoning.
Similarity (advanced): Area/volume ratio problems requiring the scale factor found indirectly.

Grades 4–5

Basic probability: Single events; listing outcomes; probability from a table.
Relative frequency: Estimate probability from experimental data.
Two-way tables: Complete and interpret; use to find probabilities.
Averages & range: Mean, median, mode, range from a list or simple frequency table.

Grades 6–7

Tree diagrams (independent): Multiply along branches, add across; "at least one" problems.
Venn diagrams: Set notation (∩, ∪, A′); find probabilities from a diagram.
Grouped data averages: Estimate the mean; find the modal/median class.
Scatter graphs: Line of best fit; describe correlation; interpolate/extrapolate.
Cumulative frequency (basic): Draw a graph; read off median and quartiles.

Grades 8–9

Tree diagrams (conditional): Dependent events (without replacement); conditional notation P(A|B).
Histograms: Frequency density with unequal class widths.
Box plots & CF (comparative): Compare distributions; interquartile range analysis.
Sampling & bias: Evaluate sampling methods for bias; suggest improvements.
Complex Venn probability: Multi-set problems requiring algebraic setup.

Grade bands are indicative — based on typical question-targeting and examiner reports, not an official Pearson document. A topic can appear at more than one grade depending on context and multi-step complexity; use this as a planning guide for sequencing revision, not a strict boundary.

Formulae: given vs. must-memorise

Given on the exam formula sheet

Area of a trapezium; volume of a prism
Compound interest formula
Quadratic formula
Sine rule, cosine rule, area = ½ab sin C
Volume & surface area of sphere, cone, pyramid

Must be memorised

Pythagoras' theorem a²+b²=c²
Trig ratios (SOHCAHTOA)
Circumference/area of a circle
Straight line y = mx + c; gradient formula
Circle theorems; vector rules
Completing the square; difference of two squares

Recent Higher tier grade boundaries

Raw marks out of 240 · June series · Pearson Edexcel official data
Series	G9	G8	G7	G6	G5	G4	G3
June 2019	198	167	137	108	80	52	38
June 2022	194	165	137	104	71	38	21
June 2023	203	174	145	112	79	47	31
June 2024	197	167	137	105	73	42	26
June 2025	217	186	156	121	87	53	36

2020–2021 are omitted as exams were cancelled/disrupted. Boundaries are set annually via Ofqual's comparable-outcomes process and vary year to year — they're a guide to the standard required, not a fixed target.

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